AXIOMATICS AND THE DEVELOPMENT OF CREATIVE TALENT

Wilder, R.L. "Axiomatics and the development of creative talent," The Axiomatic Method with Special Reference to Geometry and Physics, L. Henken,P. Suppes, and A. Tarski (eds), North-Holland, (1959), 474-488.

R.L. WILDERUniversity of Michigan, Ann Arbor, Michigan, U.S.A.

Introduction. Perhaps I should apologize for presenting here a paper that embodies no new results of research in axiomatics. However, for some time I have felt that someone should record a description of an important method of teaching based on the axiomatic method, and this conference seems an appropriate place for it.

Actually, I can point to an excellent precedent in that the late E.H. Moore devoted most of his retiring address [2], as president of the American Mathematical Society, to a Study of the role of the then rapidly developing abstract character of pure mathematics, especially the increasing use of axiomatics, in the teaching of mathematics in the primary and secondary schools. Just how much influence E.H. Moore's ideas had on the later developments in elementary mathematical education in this country, I do not know. It is perhaps significant that the increasing concern with these matters on the part of a large section of the membership of the American Mathematical Society (particularly in the Chicago Section) led, several years later, to the forming of a new organization, the Mathematical Association of America, whose special concern was with the teaching of mathematics in the undergraduate colleges.1

Historical Development of the Method. We have heard a great deal, the past fifty years or so, of the use of the axiomatic method as a tool for research. Indeed, this use of the method has been justly considered as one of the most outstanding and surprising phenomena in the evolution of modern mathematics. Scarcely a half century ago, so great a mathematician as Poincaré could devote, in an article entitled The Future of Mathematics [6], less than half a page to the axiomatic method. And although conceding the brilliance of Hilbert's use of the method, he predicted that the problem of providing axiomatic foundations for various fields of mathematics would be very "restricted", and that "there would be nothing more to do when the inventory should be ended, which could not take long. But when", he continued, "we shall have enumerated all, there will be many ways of classifying all; a good librarian always finds something to do, and each new classification will be instructive for the philosopher."

As recently as 1931, Hermann Weyl matched the contempt veiled in these remarks by a fear expressed as follows: "--I should not pass over in silence the fact that today the feeling among mathematicians is beginning to spread that the fertility of these abstracting methods [as embodied in axiomatics] is approaching exhaustion. The case is this: that all these nice general notions do not fall into our laps by themselves. But definite concrete problems were conquered in their undivided complexity, single handed by brute force, so to speak. Only afterwards the axiomaticians came along and stated: Instead of breaking in the door with all your might and bruising your hands, you should have constructed such and such a key of skill, and by it you would have been able to open the door quite smoothly. But they can construct the key only because they are able, after the breaking in was successful, to study the lock from within and without. Before you can generalize, formalize and axiomatize, there must be a mathematical substance. I think that the mathematical substance in the formalizing of which we have trained ourselves during the last decades, becomes gradually exhausted. And so I foresee that the generation now rising wi11 have a hard time in mathematics." 2

Evidently mathematical genius does not correlate well with the gift of prophecy, since neither Poincarés disdain nor Weyl's fears have been justified. Neither of these eminent gentlemen seems to have realized that a powerful creative tool was being developed in the new uses of the axiomatic method. It was Weyl's good fortune to live to see and acknowledge the triumphs of the method. And undoubtedly had Poincaré lived to observe how the method contributed to the progress of mathematics, he would gladly have admitted his prophetical shortcomings. It is easy to comprehend why they felt as they did, and as, conceivably, a majority of their colleagues felt. For until quite recent years, the method had achieved its most notable successes in geometry, where axiom systems often served as suitable embalming devices in which to wrap up theories already worked out and in a stage of decline. The value of the method as a tool for opening up vast new domains for mathematical investigation, as it has done in algebra and topology for example, was not yet sufficiently exemplified to make an impression on the mathematical public. Peano's fundamental researches in logic and number theory were concealed in his unique "pasigraphy"; and besides, was not this again a case of wrapping old facts in new dress (mused the uncomprehending analyst)? Similarly Grassmann's earlier work in his (now justly appreciated) Ausdehnungslehre was concealed in a mass of philosophical obscurities, and moreover the philosophy of the time was dominated by a Kantian intuitionism not receptive to the idea of mathematics as a science of formal structures.

Nevertheless, the evolution of modern mathematics was proceeding in a direction which made inevitable those uses of axiomatics with which every modern mathematician is now familiar. No one, among the mathematicians active around the turn of the century, appears to have been more aware of this trend than the American mathematician E.H. Moore. Moore's interest in, and use of, axiomatic procedures is well known, and I have already remarked on his interest in the influence which they might have on the teaching of elementary mathematics. Of importance for my purposes is the influence of his ideas on a group of young mathematicians who were under his tutelage at the time, particularly R.L. Moore and O. Veblen. Both Veblen and R.L. Moore wrote their doctoral dissertations in the axiomatic foundations of geometry. And the interests of both soon turned to what was at the time a new branch of geometry in which metric ideas play no official role, viz. topology, or as it was then called, analysis situs.

It is an interesting fact, however, that the topological interests of the two diverged, the one, Veblen, following the line initiated by Poincaré and subsequently called "combinatorial topology", the other, R.L. Moore, following the line stemming from the work of Cantor and Schoenflies and subsequently called "set-theoretic topology". And whereas the latter, set-theoretic topology, lent itself naturally to the axiomatic approach which Moore continued to develop, the former, combinatorial topology, was not left by Poincaré (whose feelings toward the axiomatic method we have already indicated above) in a form suitable to axiomatic development.

The first major work of R.L. Moore in "analysis situs" [3], was published in 1916.3 It embodied a set of axioms characterizing the analysis situs of the euclidean plane. In a later paper [4], Moore showed this axiom system to be categorical, and still later [5] applied it in a way prophetic of the new, creative uses of the axiomatic method soon to come into vogue.

However, of much more importance for my present purposes, was the manner in which Moore 4 used his axiom system for plane analysis situs for discovering and developing creative talent. Those of us who are accustomed to the use of axioms in constructing new theories, or for other technical creative purposes, may have lost sight of the fact that the axiomatic method can serve as the basis for a most useful teaching device.

I am not referring to the traditional use of axioms in teaching high school geometry of the euclidean type. Although here, in the hands of an inspired teacher, the method can and sometimes undoubtedly does turn up potential mathematicians, most of the teaching of high school geometry seems to be of two kinds. Either it is based on the use of a standard text book in which the theorems are all worked out in detail for study by the pupil, with a supply of minor problems -- so-called "originals" -- to be done by the pupil and geared usually to the ability of the "average" student; or it is carried out in connection with a laboratory process which is supposed to exemplify the so-called "reality" of the theorems proved, thereby preventing the abstract character of the system from becoming too dominant. In short, the whole process may be considered overly adapted to the capacities of the "average" student and consequently generally loses -- perhaps justifiably -- its potentiality for developing the mathematical talents of the more gifted student.

Nor am I referring to the fact that quite commonly, in our graduate courses in algebra and topology, we use the axiomatic method for setting up abstract systems. I mean something more than this. What I mean can perhaps be indicated by a remark which one of my former students made to me in a recent letter: "I am having quite good success teaching a course, called Foundations of Analysis, by the Moore-Socrates method." The use by Moore of the axioms for plane analysis situs in his teaching had many elements in common with the Socratic method as revealed in the "Dialogues", especially in the general type of interplay between master and pupil.

Moore proceeded thusly: He set up a course which he called "Foundations of Mathematics", and admitted to attendance in the course only such students as he considered mature enough and sufficiently sympathetic with the aims of the course to profit thereby. It was not, then, a required course, nor was it open to any and all students who wanted to "learn something about" Moore's work. He based his selection of students, from those applying for admission, on either previous contacts (usually in prior courses) or (in the case of students newly arrived on the campus) on analysis via personal interview -- usually the former (that is, previous contacts). The amazing success of the course was no doubt in some measure due to this selection process.

He started the course with an informal lecture in which he supplied some explanation of the role to be played by the undefined terms and axioms. But he gave very little intuitive material -- in fact only meagre indication of what "point" and "region" (the undefined terms) might refer to in the possible interpretations of the axioms. He might take a piece of paper, tear off a small section, and remark "Maybe that's a region". However, as the course progressed, more intuitive material was introduced, oftentimes by means of figures or designs set up by the students themselves.

The axioms were eight 5 in number, but of these be gave only two or three to start with; enough to prove the first few theorems. The remaining axioms would be introduced as their need became evident. He also stated, without proof, the first few theorems, and asked the class to prepare proofs of them for the next session.

In the second meeting of the class the fun usually began. A proof of Theorem 1 would be called for by asking for volunteers. If a valid proof was given, another proof different from the first might be offered. In any case, the chances were favorable that in the course of demonstrating one of the theorems that had been assigned, someone would use faulty logic or appeal to a hastily built-up intuition that was not substantiated by the axioms.

I shall not bore you with all the details; you can use your imaginations, if you will, regarding the subsequent course of events. Suffice it to say that the course continued to run in this way, with Moore supplying theorems (and further axioms as needed) and the class supplying proofs. I could give you many interesting -- and amusing -- accounts of the byplay between teacher and students, as well as between the students themselves; good-natured "heckling" was encouraged. However, the point to be emphasized is that Moore put the students entirely on their own resources so far as supplying proofs was concerned. Moreover, there was no attempt to cater to the capacities of the "average" student; rather was the pace set by the most talented in the class.

Now I grant that there seems to be nothing sensational about this. Surely others have independently initiated some such scheme of teaching.6 The noteworthy fact about Moore's work is that he began finding the capacity for mathematical creativeness where no one suspected it existed! In short, he found and developed creative talent. I think there is no question but that this was in large measure due to the fact that the student felt that he was being "let in" on the management and handling of the material. He was afforded a chance to experience the thrill of creating mathematical concepts and to glimpse the inherent beauty of mathematics, without having any of the rigor omitted in order to ease the process. And in their turn, when they went forth to become teachers, these students later used a similar scheme. True, they met with varying success -- after ail, a pedagogical system, no matter how well conceived, must be operated by a good teacher. Their success was striking enough, however, that one began to hear comments and queries about the "Moore method". And it is partly in response to these that I am talking about the subject today. It seems that it is time someone described the method as it really operated, and perhaps thereby cleared up some of the folklore concerning it.

Description of the Method. In the interest of clarity, I shall arrange my remarks with reference to certain items which I think, after analyzing the method, are in considerable measure basic to its success. These items are as follows:

Selection of students capable (as much as one can tell from personal contacts or history) of coping with the type of material to be studied.

Control of the size of the group participating; from four to eight students probably the best number.

Injection of the proper amount of intuitive material, as an aid in the construction of proofs.

Insistence on rigorous proof, by the students themselves, in accordance with the ideal type of axiomatic development.

Encouragement of a good-natured competition; it can happen that as many different proofs of a theorem will be given as there are students in the class.

Emphasis on method, not on subject matter. The amount of subject matter covered varies with the size of class and the quality of the individual students.

I think these six items lie at the heart of the method. Of course they slight the details; e.g., the manner in which Moore exploited the competition between students, and the way in which he would encourage a student who seemed to have the germ of an idea, or put to silence one who loudly proclaimed the possession of an idea which upon examination proved vacuous. I imagine that it was in such things as these that Moore most resembled Socrates. But these are matters closely related to Moore's personality and capability as a teacher, so I shall confine myself to the six points enumerated above so far as the description of the method is concerned. They are, I realize, themselves pedagogic in nature, but more of the nature of what I might call axiomatic pedagogy. They constitute, I believe, a guide to the successful use of axiomatics in the development of creative talent.

I would like to comment further on them:

1. Selection of students capable of coping with the type of material to be studied. I have already made some remarks in this regard. I pointed out that Moore based his judgment regarding maturity either on his experience with the student in prior courses, or on personal interviews. I might add, parenthetically, that as the years went by and his students began to use his methods in their own teaching, a sort of code developed between them whereby one of the "cognoscenti" would apprise one of his colleagues in another university of the availability of potentially creative material. For example, the "pons asinorum" of Moore's original axiom system was "Theorem 15". If one of Moore's graduates wished to place a student for further work under the tutelage of another of Moore's students at a different institution, and could include in his recommendation the statement, "He proved Theorem 15", then this became a virtual "open sesame".

But Moore, himself, was not dependent on other institutions; he found his students, generally speaking, in the student body of The University of Texas. He had a singular ability for detecting talent among undergraduates, and often set his sights on a man long before he was ready for graduate work. Indeed, in some instances, he would allow in his class in "Foundations" an undergraduate whom he deemed ready for creative work. For Moore believed that a man should start his creative work as soon as possible, and the younger the better. He reasoned that one could always pick up "breadth" as he progressed. It was not unusual for him to discover talent in his calculus classes. And once he suspected a man of having a potentially mathematical mind, he marked that man for the rest of the course as one with whom he would cross his foils, so to speak. By the end of the term, he was usually pretty sure of his opinion of the man.

Of course he could not, in the very nature of the case, always be certain. This applies especially to those who entered his class as graduate students from other institutions, who had had no previous work with him, and whom he had to screen usually in a single interview at registration time. And when a student of little or no talent did slip by, he was doomed to a semester of either sitting and listening (usually with little comprehension), or to feverishly taking notes which he hoped to be able to understand by reading outside of class. In the latter case he was often disappointed, for as we all know, one's first proof of a theorem is usually not elegant, to understate the case, and the first proofs of a theorem given in class were likely to be of this kind. But as I stated before, the aim of the course was not so much to give certain material -- the student who wished the latter would have been better advised to read a book or to seek out the original material in journals. I would call these "note-takers" the "casualties" of the course. So you see it was humane, as well as good strategy, to allow only the "fit" to enroll in the course.

I might remark, too, that those of us who went from Texas to other institutions as young instructors, did not usually find it possible to institute Moore's "exclusion policy" in all its rigor. For various reasons, we often had to throw our courses open to one and all. This naturally led to certain modifications, as, for instance, making sure that the "note takers" ultimately secured an elegant proof; this seemed the least that they were entitled to under a system where they were not sufficiently forewarned of what to expect, and of especial importance if the material covered was to be used by the student as basic information in later courses.

2. Control of the size of the group participating; from four to eight probably the best number. This is obviously not independent of 1., since Moore's method of selecting students was clearly suited to keeping down the size of the class. Some of us, however, especially during periods of high enrollments, have had to cope with classes of as many as 30 students or more. I can report from experience that even with a class this large, the method can be used. Or course inevitably a few (sometimes only two or three) students "star in the production". I have found, however, that these "star" students often profited from having such a large audience as was afforded by the "non-active" portion of the class. Often the "nonstars" came up with some good questions and sometimes -- rarely to be sure -- with a suggestion that led to startling consequences.

In short, although from four to eight is the ideal size of class for the use of the axiomatic method, it is not impossible to handle classes of as many as 30 while using the method.

3. Injection of the proper amount of intuitive material, as an aid in the construction of proofs. This, I hardly need emphasize, must be handled carefully. With no intuitive background at all, the student has little upon which to fix his imagination. The undefined terms and the axioms become truly meaningless, and a mental block perhaps ensues. Here the instructor must exercise real ingenuity, striving to furnish that amount of intuitive sense that will be sufficient to suggest processes of proof; while at the same time holding the student to the axiomatic basis as a foundation for all assertions of the proof.

I have always been interested, in my use of the axiomatic method in Topology, in observing the degree to which the various students used figures in giving a demonstration. Some relied heavily on figures; others used none at all, being content to set down the successive formulae of the proof. I have noticed that the former type of student usually developed an interest in the geometric aspects of the subject, following the tradition of classical topology, while the latter developed greater interest in the new algebraic aspects of the subject. There may be considerable truth in the old folklore that some are naturally geometric-minded, while others have not so much geometric sense but show great facility for algebraic types of thinking. I don't know of any better way to discover a student's propensities in these regards, than to give him a course in modern topology on axiomatic lines.

4. Insistence on rigorous proof, by the students themselves, in accordance with the ideal type of axiomatic development. I want to emphasize here two advantages that the axiomatic development offers.

In the first place, I have seen the method rescue potentially creative mathematicians from oblivion. Without knowing the reason therefore, they had become discouraged and depressed, having taken course after course without "catching on" -- with no spark of enlightenment. The reason for this was evidently that their innate desire for clearcut understanding and rigor was continually starved in course after course. One can appreciate the gleam in a student's eye when, provided with the type of rigor which the axiomatic method affords, he finds his mathematical self at last; for the first time, seemingly, he can let his creative powers soar with a feeling of security. This is truly one of the ways in which creative talent is discovered.

In the second place, even the average student feels happy about knowing just what he is allowed to assume, and in the feeling that at last what he is doing has, in his eyes, an almost perfect degree of validity. I can illustrate by an example here. I was once giving a course in the structure of the real number system, using a system of axioms and the "Moore method". In the class was a man who had virtually completed his graduate work and was writing his dissertation in the field of analytic functions; At the end of the course he came to me and said, "You know, I feel now for the first time in my life, that I really understand the theory of real functions". I knew what had happened to him. Despite all his courses and reading in function theory, he had never felt quite at ease in the domain of real numbers. Now he felt that, having been thrown wholly on his own resources, he had come to grips with the most fundamental properties of the real number system, and could, so to speak, "look a set of real numbers in the eye!"

5. Encouragement of a good-natured competition. I have found that an interesting by-play often developed between students, either to see who could first obtain the proof of a theorem; or failing that, who could give the most elegant proof. I presume this is a foretaste of the situation in which the seasoned mathematician often finds himself. I hardly need to cite historic instances to an audience like this; instances in which a settlement of a long outstanding problem was clearly in the offing, and the experts were vying with one another to see who would be the first to achieve the solution. This always adds zest to the game of mathematics, either on the elementary level or on the professional level. And no system of teaching lends itself better to this sort of thing than the one I am discussing.

There is also the possibility that an original-minded student will discover a new and more elegant proof of a classical theorem. I have had this happen several times, and on at least one occasion, to which I shall refer again below, the proof given failed to use one of the conditions stated in the classical hypothesis, so that a new and stronger theorem resulted.

6. Emphasis on method, not on subject matter. When one lectures, or uses a text, the student is frequently presented with a theorem and then given its proof before he has had time to digest the full meaning of the theorem. And by the time he has struggled through the proof presented, he has been utterly prejudiced in favor of the methods used. They are all that will occur to him, as a rule. Use of the axiomatic method with the student providing his own proof, forces an acquaintance with the meaning of the theorem, and a decision on a method of proof. I have continually in my classes, whenever existence proofs were demanded, urged the students to find constructive methods whenever possible. In this way, I have had presented to me constructive proofs in instances where I did not theretofore know that such proofs could be given.

In short, use of the axiomatic method not only encourages the student to develop his own creative powers, but sometimes leads to the invention of new methods not previously conceived.

There is one other feature of the method as Moore used it that I have omitted above, for various reasons, chiefly because of the vagueness of its terms and the debatability of any interpretation of it:

7. Selection of material best suited to the method. It is probably wisest to select certain special subjects which seem best suited for the avowed purposes of discovering and developing creative ability. For example, one might select material that presupposes little in the way of special techniques (as, for instance, the techniques of classical analysis), but that does require that ability to think abstractly which should be a characteristic of the mature student of mathematics, and which requires little intuitive background. The material which Moore chose was of this nature; another such selection might be the theory of the linear continuum.

In the case of the material which Moore selected, the student was led quickly to the frontiers of knowledge; that is, to the point where he might soon be doing original research. I think this aspect of his method is not, however, essential to its success in developing creative talent. As Moore used the method, the line between what was known and what was unknown was not revealed to the students. Customarily they were not apprised of the source of the axioms or the theorems; for all they knew, these had probably never been published. And he could go on with them to unsolved problems through the device of continuing to state theorems whose validity he might not himself have settled, without their ever being aware of the fact.

Consequently, so far as this item 7 is concerned, I would say that the important aspect of it is the selection of material requiring little intuitive background and presupposing mathematical maturity but little technique. The techniques of deduction, proof, and of discovering new theorems are naturally part of the design of the course; the axiomatic method is ideal for the development of these, and they should be given priority over the quantity of material covered.

The justification for the system is of course its success. It soon reveals to both teacher and student whether or not the latter possesses mathematical talent. It quickly selects those who have the "gift", so to speak, and develops their creative powers in a way that no other method ever succeeded in doing. Every mathematician, now and in the past, has recognized the necessity for doing mathematics, not just reading it, and has assigned "originals" for the student to do on his own. In the Moore system, we find the "original" par excellence -- there is nothing in the course but originals! I should repeat, in connection with these remarks, that it is not unusual for a student to find a new proof of a known theorem that deserves publication, as well as for new theorems to be found. I had one outstanding case of this in my own use of the method, where the new proof showed one could dispense with part of the traditional hypothesis; and I had the student go on to incorporate his methods into proving another and similar theorem which was historically related to the former and was susceptible to the same improvement in its hypothesis.

The fact that what the logician would call the "naive" axiomatic method is used, does not seem to cause any objection from the student. In fact, I am afraid that a strict formalism might not work so well; although this is debatable, and certainly a carefully formulated proof theory would be quite adaptable to certain types of material. The use of a "natural" language throughout, except for the technical undefined terms, was, however, an important feature of the method as Moore used it, not only aiding the intuition but enabling that competition mentioned in item 5 to "wax hot" at crucial moments.

This brings me to some remarks about an area of teaching in which tradition is most strong, namely the undergraduate curriculum. Today we hear a great deal about encouraging the young student to go into a mathematical or scientific career. Unfortunately much potential creative talent is lost to mathematics early in the undergraduate training, and much of this, I am sure, is due to traditional modes of presentation. It is possible that the axiomatic approach offers at least a partial solution of this problem.

The axiomatic method in, the undergraduate course. As Moore used the axiomatic method for teaching on the graduate level, the aim was to discover and develop creative ability. Is there not a possibility that the method could be employed to advantage at a lower level, so that the potentially creative mathematician will be encouraged to continue in mathematics to the point where his talents can be more decisively put to the test?

I am convinced that one of our greatest errors in the United States educational system has been to underestimate the ability of the young student to think abstractly. Moreover, I am convinced that as a result, we actually force him to think "realistically" where actually he would prefer to think abstractly, so that by the time he begins graduate work, his ability to abstract has been so dulled that we have to try to develop it anew.

It seems probable that we could try using the axiomatic method on a lower level, perhaps even on the freshman level, at selected points where the material is of a suitable kind. In the interests of caution, perhaps we should experiment on picked groups first, as well as with carefully selected material. It is possible that we might light creative sparks where, with the conventional type of teaching, no light would ever dawn. Some years ago I had a chance to do this sort of thing, with a picked group of around ten students. I did not have an opportunity to teach most of these students again until they became graduates. But I am happy to state that a majority of them went on to the doctorate -- not necessarily in mathematics, for some turned to physics -- but at least they went on into creative work. I don't wish to give myself credit here; it is the method that deserves the credit. These men discovered unsuspected powers in themselves, and could not resist cultivating and exercising them further. Moreover, I found they were delighted at being able to establish their ideas on a rigorous basis. For example, in starting the calculus, I gave them precise definitions, etc., for a foundation of the theory of limits in the real number system, and let them establish rigorously on this foundation all the properties of limits needed in the calculus. The result was that they covered the calculus in about half the time ordinarily required. Admittedly some of this saving in time was due to the select nature of the class, but a major part, I am convinced, was due to the confidence and interest induced by establishing the theory of limits on firm basis.

In the presidential address of E.H. Moore to which I referred in my introduction, he stressed the advisability of mixing the real and the abstract in the teaching of mathematics in the secondary schools. But (and here I quote from E.H. Moore's address, p. 416) "-- when it comes to the beginning of the more formal deductive geometry why should not the students be directed each for himself to set forth a body of geometric fundamental principles, on which he would proceed to erect his geometrical edifice? This method would be thoroughly practical and at the same time thoroughly scientific. The various students would have different systems of axioms, and the discussion thus arising naturally would make clearer in the minds of all precisely what are the functions of the axioms in the theory of geometry." Here was evidently a suggestion for the creative use of axiomatics at the high school level.

There are currently experiments being conducted in some undergraduate colleges which are based on modifications of the methods Moore used. For example, I know of one case 7 where a special course of this kind, for freshmen, has been devised. One-half the course is spent establishing arithmetic, on an axiomatic basis. The numbers O, 1, 2, etc, are used, but the development is rigorous, and indeed approaches the rigor of a formal system in that the rules for proof are explicitly set forth. By the use of variables, the student is led gradually into algebra, which occupies most of the latter half of the course. The course terminates in an analysis, based on truth tables, of the formal logic to which the student has gradually become accustomed during the course. I judge that one of the reasons for the success which the course seems to have achieved is that the student is made aware of the reasons for the various arithmetic manipulations in which he was disciplined in the elementary schools; as, for instance, why one inverts and multiplies in order to divide by a fraction. This course has, incidentally, revealed that students who do not do well on their placement examinations are not necessarily laggards, weak-minded, or susceptible of any of the other easy explanations, but that they often are intelligent, capable persons who have been antagonized by traditional drill methods. Moreover, some of these students are induced by the course into going further in mathematics. I believe this course is still in a developmental stage, and I await with interest reports on its effectiveness. One gets the feeling from reading the text used that the student is being treated with trust, as naturally curious to know the why of what he is doing, and as being intelligent enough to find out if permitted!

During the past few years there has been published a number of elementary texts which use the axiomatic method to some extent. Perhaps this is a sign of a trend. I hope that in my remarks I have not overemphasized to such an extent as to give an impression that I think the axiomatic method is a cure-all. I do not think so. Nor do I think it desirable that all courses should be axiomatized! But I believe that the great advances that the method has made in mathematical research during the past 50 years can, to a considerable extent, find a parallel in the teaching of mathematics, and that its wise and strategic use, at special times along the line from elementary teaching to the first contacts with the frontiers of mathematics, will result in the discovery and development of much creative talent that is now lost to mathematics.

1 See [1], parts VII and XV 6, but especially p. 81 and p. 146.2 Quoted from H. Weyl [7], It is to Weyl's credit that he acknowledges, in this connection, the brilliant results obtained by Emmy Noether by her pioneering use of the axiomatic method in algebra.3 There are three axiom systems given in this paper. In our remarks we refer only to that one which is designated in [3] by the symbol 1.4 From here on, by "Moore" I shall mean R.L. Moore.5 One of these was later shown (by the present author [8] not to be independent.6 Professor A. Tarski informed me after the reading of this paper that he had used a somewhat analogous method in one of his courses at the University of Warsaw.7 At the University of Miami.